| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Topic | Fixed Point Iteration |
| Type | Derive stationary point equation |
| Difficulty | Moderate -0.3 This is a structured multi-part question covering standard A-level techniques: finding roots by setting factors to zero (trivial), differentiating a product and rearranging (routine), sign change verification (direct substitution), and applying a given iterative formula (mechanical calculation). While it requires several steps, each part follows predictable patterns with no novel problem-solving or insight required. Slightly easier than average due to the highly scaffolded nature. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ace492d8-1dd0-401e-af74-505ca19d5e9c-20_679_1136_132_566}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where
$$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$
The curve cuts the $x$-axis at the points $A$ and $B$ and has a maximum turning point at $Q$, as shown in Figure 2.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$ coordinate of $A$ and the $x$ coordinate of $B$.
\item Show that the $x$ coordinate of $Q$ satisfies
$$x = \frac { 8 } { 1 + \ln x }$$
\item Show that the $x$ coordinate of $Q$ lies between 3.5 and 3.6
\item Use the iterative formula
$$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$
with $x _ { 1 } = 3.5$ to
\begin{enumerate}[label=(\roman*)]
\item find the value of $x _ { 5 }$ to 4 decimal places,
\item find the $x$ coordinate of $Q$ accurate to 2 decimal places.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q9 [9]}}